Examples of Isomorphic Real and Complex Lie Groups Are there any examples of common (for example SL, SO, SU, GL groups of varying dimension) real and complex Lie groups which are isomorphic? I'm aware that many Lie groups are of odd dimension which would stop an argument being made by taking a real manifold and showing it's equivalent to a complex one but writing about some of these groups (e.g. SO and SU) often it isn't explicitly stated whether they are real or complex Lie groups.
In the case of $GL(\mathbb{C},n)$ and $GL(\mathbb{R},n)$ as the former is of dimension $2n^2$ and the latter $n^2$ assume no GL groups can be isomorphic as they cannot have the same dimension. I'm not sure however if dimension stops something being isomorphic as a group (certainly they are not isomorphic as manifolds).
 A: As noticed in comments, this question is a bit too vague and the answer is, in one way, simple ("well, look at all complex Lie groups you know and check what real Lie group you get when you forget the complex structure"), and in one way difficult (because now you would need to know all real Lie groups up to isomorphism, which are ... many).
Because of your insistence on some author's flaw to write, for example, $SO_n$ and to not make it sufficiently clear whether they mean the complex Lie group $SO_n(\mathbb C)$ or the real Lie group $SO_n(\mathbb R)$, it just occurred to me that you might ask if that ever does not make a big difference, i.e. whether one ever has an isomorphism of underlying real Lie groups like $$SO_n(\mathbb R) \stackrel{?}\simeq SO_n(\mathbb C)$$ or $$GL_n(\mathbb R) \stackrel{?}\simeq GL_n(\mathbb C)$$ or $$SL_n(\mathbb R) \stackrel{?}\simeq SL_n(\mathbb C)$$
with the same algebraic group and the same $n$ on both sides?? Well, with possible trivial exceptions for $n=1$ in some cases, easy dimension considerations should rule out that any of such isomorphisms would ever exist, even on the manifold level. So no, if an author is imprecise there, it's really their fault.
The question becomes interesting if one allows for a little more vagueness, and asks for possible "exceptional" isomorphism (cf. https://mathoverflow.net/q/81344/27465) with a complex Lie group on the one side and a classical real Lie group on the other. I.e., can something like $SO_{7}(\mathbb C) \stackrel{?}\simeq SL_{12}(\mathbb R)$ ever happen? Well, the first (but only) standard example is
$$SL_2(\mathbb C) \simeq Spin(1,3)$$
(as real Lie groups), i.e. forgetting the complex structure on the left hand side gives us a real Spin group (which happens to be the double cover of the indefinite special orthogonal group $SO(1,3)$, all this higly related to the Lorentz group and thus of interest to physics). In particular as real Lie algebras,
$$\mathfrak{sl}_2(\mathbb C) \simeq \mathfrak{so}(1,3).$$
Actually, on the Lie algebra level the question will be generally answerable in the semisimple case, as the Satake-Tits diagram of the underlying real Lie algebra of any semisimple complex Lie algebra just consists of two copies of the Dynkin diagram of the complex one, with arrow between them. With the exception of the above (where $A_1 \times A_1$ happens to be $D_2$, and the arrow between these two unconnected nodes tells us we are in a quasi-split case), none of these "two copies of a Dynkin diagram connected with arrows" have a different "classical" name, as far as I know.
